Periodic Patterns and Energy States of Buckled Films on Compliant Substrates

Transcription

1 Periodic Patterns and Energy tates o Buckled Films on ompliant ubstrates The Harvard community has made this article openly available. Please share how this access beneits you. Your story matters. itation Published Version Accessed itable Link Terms o Use ai, hengqiang, Derek Breid, Alred J. rosby, Zhigang uo, and John W. Hutchinson.. Periodic patterns and energy states o buckled ilms on compliant substrates. Journal o the Mechanics and Physics o olids 59(5): doi:.6/j.jmps... August 6, 8 :: AM EDT This article was downloaded rom Harvard University's DAH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set orth at (Article begins on next page)

2 Periodic Patterns and Energy tates o Buckled Films on ompliant ubstrates. ai*, D. Breid**, A. J. rosby**, Z. uo* and J.W. Hutchinson* chool o Engineering and Applied ciences* Harvard University, ambridge, MA 38 Polymer cience and Engineering** University o Massachusetts, Amherst, MA Abstract. Thin sti ilms on compliant elastic substrates subject to equi-biaxial compressive stress states are observed to buckle into various periodic mode patterns including checkerboard, hexagonal and herringbone. An experimental setting in which these modes are observed and evolve is described. The modes are characterized and ranked by the extent to which they reduce the elastic energy o the ilm-substrate system relative to that o the unbuckled state over a wide range o overstress. A new mode is identiied and analyzed having nodal lines coincident with an equilateral triangular pattern. Two methods are employed to ascertain the energy in the buckled state: an analytical upper-bound method and a ull numerical analysis. The upper-bound is shown to be reasonably accurate to large levels o overstress. For lat ilms, except at small states o overstress where the checkerboard is preerred, the herringbone mode has the lowest energy, ollowed by the checkerboard, with the hexagonal, triangular and onedimensional modes lowering the energy the least. At low overstress, the hexagonal mode is observed in the experiments not the square mode. It is proposed that a slight initial curvature o the ilm may play role in selecting the hexagonal pattern accompanied by a detailed analysis. An intriguing inding is that the hexagonal and triangular modes have the same energy in the buckled state and, moreover, a continuous transition between these modes exists involving a linear combination o the two modes with no change in energy. Experimental observations o various periodic modes are discussed with reerence to the energy landscape. Discrepancies between observations and theory are identiied and open issues are highlighted. Keywords: Thin ilms, compliant substrates, buckling, wrinkling, mode transitions

3 . Introduction Due to processing, thermal expansion mismatch, or dierential expansion due moisture changes, thin metal or ceramic ilms on elastomer or polymer substrates oten experience equi-biaxial in-plane compression and buckle into intriguing periodic mode patterns (e.g., Bowden et al., 998; Huck et al., ; Breid and rosby, 9). Examples o one-dimensional, square checkerboard, hexagonal, triangular and herringbone modes are depicted in Fig., and experimental observations o several modes rom the work o Breid and rosby (9, ) are shown in Fig.. Motivated by experimental observations presented in the next section, this paper builds on the theoretical studies o hen and Hutchinson (4) and Audoly and Boudaoud (8 a,b,c) that have elucidated nonlinear aspects o the buckling behavior o some o the periodic modes. In the range o moderate to large overstress, hen and Hutchinson (4) showed that, among one-dimensional (straight-sided), square checkerboard and herringbone modes, the herringbone mode has the lowest energy in the buckled state while the one-dimensional mode has the greatest. Audoly and Boudaoud (8a) examined urther details o the post-buckling behavior o these modes, including the range in which the one-dimensional mode is stable and its transition to the herringbone mode under stress states which are not equi-biaxial. They also considered the hexagonal mode and they showed that the square mode has the lowest energy in the range o small overstress. In companion papers, Audoly and Boudaoud (8 b,c) used asymptotic methods to explore aspects o behavior expected in the range o very large overstress with emphasis on the herringbone mode. In this paper, we employ an analytical upper-bound calculation and numerical inite element analysis to explore urther aspects o the nonlinear post-buckling behavior o the ilm/substrate system with the aim o shedding light on the relation between observed periodic patterns and the level o overstress noted in the next section. Behavior over a wide range o stress rom the onset o buckling to advanced post-buckling at large overstress is investigated or ilms under equi-biaxial stress in the unbuckled state. The problem in which the pre-buckling stress state is equi-biaxial compression is unusual in that there are a multiplicity o periodic modes associated with the critical buckling stress, including one-dimensional (D), checkerboard and hexagonal modes and a newly

4 identiied triangular mode. The initial and advanced post-buckling behavior o all these modes and the relations among them are studied here. In addition, results are obtained or a herringbone mode which has the lowest energy o all the modes considered at suiciently large overstress. Because the herringbone mode is not one o the modes associated with the onset o buckling, it is not avored at small overstress. The model investigated analytically is a nonlinear von Karman plate bonded to a linear elastic oundation; it is introduced in ection 3. A three-dimensional model o the ilm and the substrate is employed in the numerical inite element simulations. For the analytical model, the issue o the most appropriate traction conditions linking the plate to the oundation, which was addressed by Audoly and Boudaoud (8a), is considered here in more detail. The various modes are introduced in ection 4. Upper-bounds to the energy states are obtained in ection 5, supplemented by results rom a inite element analysis o the three-dimensional model given in ection 7. The role o an initial spherical curvature o the ilm is addressed in ection 6 in an eort to explain some o the apparent conlicts between theory and experimental observation identiied in the next section.. Experimental observations o mode patterns motivating theoretical issues The relation between the buckling modes and overstress has been studied experimentally or crosslinked polydimethylsiloxane (PDM) substrates ( E MPa,.5, thickness ~ 3mm) whose suraces have been modiied using an ultravioletozone (UVO) oxidation process to produce surace layers, or ilms, with increased stiness relative to the underlying non-modiied elastomer substrate (Breid and rosby, 9; Breid and rosby, ). Although the ilm modulus and thickness are not known precisely, approximate values have been estimated by wrinkle wavelength measurements (han and rosby, 6) and compared with published relectivity measurements on similar materials (Eimenko et al ; Mills et al 8). The ilm thickness lies in the range rom to nm (Eimenko et al, Mills et al 8), while its modulus is o order GPa (han and rosby, 6). To induce wrinkling, the PDM samples are placed in a sealed chamber containing a reservoir o ethanol, which does not contact the samples. The ethanol vapor 3

5 is absorbed into the oxidized ilm to a greater extent than the underlying non-modiied elastomer, thus dierential swelling between the two layers leads to the development o an equi-biaxial compressive stress in the ilm. An increase in the UVO exposure time results in a thicker oxide layer with a greater swelling extent, eectively increasing the applied overstress. The applied swelling stress can be controlled alternatively by changing the vapor pressure o ethanol (Breid and rosby, ). For both cases, the magnitude o the applied overstress is determined by measuring the curvature o cantilever beams o the same composition exposed to the same vapor swelling conditions. onventional bi-layer beam bending mechanical models provide the values o the average overstress in the ilm by relating the measured curvature o the beams to the curvature measured at the critical buckling point. The magnitude o the applied strains can be estimated by measuring the applied strain at the critical buckling point or uniaxially compressed samples. Using these measurements, we estimate the total strains to range rom -%, with a critical strain around -3%. Fig. 3 collects observations rom several experiments displaying the progression o observed patterns or increasing levels o overstress. Throughout the paper, denotes the equi-biaxial stress in the ilm in the unbuckled state, is the critical value o this stress at the onset o buckling, and the extent to which / exceeds unity is reerred to as the overstress. A pattern emerges in Fig. 3 at / with an almost random array o low-amplitude undulations. We identiy the lowest ethanol vapor raction at which these undulations are observed or a given UVO treatment time as the critical buckling point, rom which the ollowing overstress values are calculated. At slightly larger overstress, /.3, a distinct hexagonal pattern is evident with central regions o all the hexagons delected into the substrate. For /.7 the herringbone pattern is dominant with occasional deects. An exceptionally well organized herringbone pattern is observed or the highest overstress shown, / 4., corresponding to the longest UVO treatment o 6 minutes. The preerence o our experimental system to assume the hexagonal mode at low overstress is overwhelming, despite the act that the square mode has been shown here and previously to have lower energy in this range when the ilm/substrate system is lat. 4

6 Furthermore, we have only observed hexagonal patterns or which the regions inside the hexagons buckle into the substrate (e.g. Figs. -4), while the theory developed later in the paper or lat ilms/substrate systems predicts that buckling o the hexagonal regions into or out o the substrate should be equally likely. One possible artiact o our experimental system is the presence o low amplitude, laterally extensive curvature o the surace o the system that exists initially or develops upon swelling in the lat UVO-PDM samples. This clue has been pursued theoretically in the paper. It will be seen that initial spherical curvature o the ilm is likely to explain the two experimental observations cited above that are otherwise inexplicable when the ilms are taken to be lat. As the applied overstress increases beyond the ormation o the hexagonal mode, a transition point to more energetically avorable herringbone patterns is observed. However, experimentally there is a tendency to maintain the hexagonal lattice o the original pattern, perhaps due to kinetic considerations o orming an entirely new pattern with dierent periodic wavelengths. The mechanism by which the hexagonal mode transitions to a more energy-minimizing pattern is seen in Fig. 4. tarting rom a pure hexagonal pattern, slight increases in the overstress cause isolated hexagons to coalesce with neighbors producing an extended local groove. The coalescing o a pair o hexagons tends to also trigger the coalescing o a neighboring pair. These triggered pairs are usually situated along a lattice line that is not parallel to that o the original pair, in order to accommodate the local stress in an equi-biaxial manner. In some cases, these coalesced grooves link to orm even longer grooves, but in general they tend to remain the product o just two or three hexagons. The overall result is a pattern that locally resembles a segmented labyrinth, or a herringbone pattern with no global orientation, analogous to the labyrinth pattern reported or homogeneously initiated wrinkling at high overstress (Huang, Hong and uo, 5; Lin and Yang, 7). In contrast, well-ordered herringbone patterns develop or systems that bypass the lower energy buckling modes or jump to high overstress values. The experimental observations noted here have motivated us to look or a theoretical explanation o why the square mode is never observed or our experimental system even though under equi-biaxial stressing it has lower energy than the hexagonal mode, assuming lat ilms. Moreover, a new triangular mode will be identiied that has 5

7 precisely the same energy in the buckled state as the hexagonal mode, and this mode has not been observed either. We also wish to explain why the hexagonal mode has always been observed to buckle with the hexagonal regions directed into the substrate, while the theory suggests there should be no such preerence. These discrepancies between theory and experiment have motivated other avenues o exploration in the theory, including the roles o initial ilm curvature and nonlinearity o the substrate. Embedded within the paper are several auxiliary indings: (i) The only modes whose nodal lines coincide with a pattern ormed by regular polygons are the equilateral triangle mode and square mode the so called hexagonal mode is ormed rom a mixture o hexagons and triangles in the manner o a Kagome pattern. (ii) Among all rectangular checkerboard modes, the square mode has the lowest energy. (iii) The hexagonal mode and triangular mode have identical energy in the buckled state, to the order obtained here, and a continuous transition exists rom one to the other at constant energy. (iv) Within the range o overstress considered in this paper, nonlinearity o the substrate has essentially no inluence on the buckling patterns. (v) A slight curvature o the ilm is likely to be playing a critical role in the mode selection observed experimentally and in determining the sign o the hexagonal delection. 3. The analytical model The ilm is described by the von Karman plate equations, and the ininitely deep substrate under the ilm is characterized by linear elasticity. Both the ilm and the substrate are taken to be isotropic and homogeneous. The process that produces the sti ilm in our experimental system almost certainly results in a modulus variation through its thickness, maintaining in-plane isotropy. An inhomogeneous ilm o this type can be replaced by a homogeneous ilm whose modulus and thickness are chosen to reproduce the bending and stretching stiness o the inhomogeneous ilm. This section begins by listing the governing equations and by recording what are now well known results or the critical stress and associated eigenmodes associated with the onset o buckling or conditions in which the ilm is stressed in equi-biaxial compression. uch conditions would arise, or example, i the substrates coeicient o thermal expansion were greater than that o the ilm and a reduction in temperature took place or i the ilm were to 6

8 attempt to swell relative to the substrate under ingress o moisture or solvent, as in the case o the experiments discussed in ection. There is some latitude in the ormulation o the model in the choice o conditions used to attach the ilm to the substrate and this issue is investigated. (3.) The governing equations Let t be the thickness o the ilm with E and as its modulus and Poisson s ratio. The substrate is taken to be ininitely deep with modulus and Poisson s ratio, E and. In artesian coordinates, ( x, x ), the von Karman plate equations representing the ilm are 4 D w ( N w, N w, N w, ) p (3.) 4 F w, w, w, Et (3.) Here, 3 D Et /[( )] is the ilm bending stiness, wx (, x ) is the vertical displacement o the ilm middle surace, and 4 () is the bi-harmonic operator. The resultant in-plane stresses in the ilm are given in terms o the airy stress unction, F, by N is F, N F, and N F,., The downward vertical traction exerted by the substrate on the delected ilm p. As is customary in this model, horizontal tractions exerted on the ilm by the substrate are neglected. For vertical delections, w wˆ cos( kx) cos( kx), p pˆ cos( k x )cos( k x ) with * described by (3.) and (3.), two choices o studies: E * pˆ wˆ / E k k. Within the ramework /( ) * E have been made in nearly all earlier E (with E E ) i tangential tractions on the substrate interace are taken to be zero (Allen (969) and many others ollowing his work), while E [4( ) / (3 4 )] E i tangential displacements at the substrate interace are * taken to be zero (Huang, 5; Audoly & Boudaoud, 8). The actual conditions (i.e. continuity o both tangential tractions and displacements across the ilm/substrate interace) require tangential tractions to be taken into account in the ilm which leads to a 7

9 more complex ormulation than that described by (3.) and (3.). This issue was addressed by Audoly & Boudaoud (8a) and will be discussed and urther clariied at the end o this section. As noted by hen and Hutchinson (4), the spread between these two estimates vanishes i the substrate is incompressible (i.e.,, i / ) and is still very small i / 3 (i.e., [4( ) / (3 4 )] [4( ) / (3 4 )] 6 /5 ). For the analytic upper-bound results presented later in ection 4, the choice or * E is let open. A three dimensional ormulation is adopted or the numerical calculations and ull continuity will be imposed across the ilm/substrate interace. Denote the equi-biaxial compressive stress in the unbuckled ilm by such that F t x x /. With L and L as the periodicity lengths in the x and x directions, supplementary compatibility conditions that ensure that the average displacements tangent to the ilm are the same as those in the unbuckled state are Et Et ( ), N N dx L dx L L E ( ), N Ndx L dx L L E w w (3.3) (3.) The critical stress and associated buckling modes The critical ilm stress,, characterizing the onset o buckling is (Allen, 969) * E E 3 4 E /3 (3.4) where E E and /( ) * E was introduced above. All modes satisying sin( kx) sin( kx) w cos( kx ) cos( kx ) * with k k k t E E /3 are eigenmodes associated with (hen and Hutchinson, 4). 3 / (3.5) (3.3) The critical buckling stress and the choice o * E 8

10 We digress to present other estimates o, including results o an exact calculation, with the purpose o guiding the choice o * E or the ormulation laid out in ection (3.). For plane strain buckling in the ( x, x ) plane, three results are presented: ase (I) Full continuity o tractions and displacements applied at the mid-plane o the von Karman plate. ase (II) Full continuity o tractions and displacements applied at the bottom surace o the von Karman plate. ase (III) The exact plane strain solution in which the ilm is treated as a prestressed elastic layer o inite thickness with ull continuity o tractions and displacements applied at the interace between the ilm layer and the substrate. elected details underlying the calculations in these cases are presented in the Appendix. The two possible choices o * E noted ection (3.), one based on zero tangential tractions acting on the substrate and the other based on zero tangential displacements imposed on the substrate, might be expected to predict critical stresses that bracket the actual critical stress. This would indeed be true i ase (I) were a correct description. This is seen in Fig. 5 or / 3 where rom ase (I) alls just below the estimate based on E [4( ) / (3 4 )] E. However, ase (II) provides a more realistic * description, and the critical stress based on ase (II) lies above the estimate based on E [4( ) / (3 4 )] E. The result o the exact calculation, ase (III), is the * uppermost curve in Fig. 5. ase (III), detailed in the Appendix, makes use o the exact ormulation o Biot (965) and Hill and Hutchinson (975) or the plane strain biurcation analysis or an incompressible pre-stressed layer o inite thickness which is coupled to the compliant linear substrate through continuity o tractions and displacements. The exact results are limited to / or the ilm, but this restriction should not be signiicant or the comparisons in Fig. 5 because does not play a role in any o the other estimates other than through E E. Fig. 6 displays the dependence o the exact predictions /( ) or various substrate Poisson ratios. As /, all the estimates based on the plate 9

11 /3 model approach ( E /4)(3 E / E ), and the exact result is only slightly greater. Many recent experiments employ elastomeric substrates which are nearly incompressible. For such systems the issue o the choice o * E vanishes. Otherwise, within the ramework o the model ection (3.) wherein only the normal traction exerted on the ilm by the substrate is taken into account, it is clearly best to choose E [4( ) / (3 4 )] E, as concluded by Audoly and Boudaoud (8a). As seen * in Fig. 5, this approximation will only underestimate the critical stress by several percent or compliant substrates with / 3 and E / E.. These conclusions apply to all o the critical modes satisying (3.5) because each o them can be expressed as a superposition o D plane strain modes. 4. The buckling modes This paper will ocus on our modes associated with the critical stress: D mode: w tcos( kx ) (4.) quare checkerboard mode: w tcos( kx )cos( kx) (4.) Hexagonal mode: w t cos( kx 3 ) cos( kx)cos( kx) (4.3) Equilateral triangular mode: w t sin( kx 3 ) sin( kx)cos( kx) (4.4) Aspects o the last three modes are depicted in Fig. 7. Each o these modes can be expressed as a superposition o critical D modes. In coordinates ( x, x ) rotated 45 o rom ( x, x ), the checkerboard mode (4.) can be re-expressed as w t cos( kx) cos( kx) /. With x ( x 3 x)/ and x ( x 3 x)/, the hexagonal mode (4.3) is w t(cos( kx ) cos( kx) cos( kx)). As noted in Fig. 7, the so- Audoly and Boudaoud (8a) use the superposition o the two D modes in their analysis o the square checkerboard mode. They incorrectly assert that hen and Hutchinson (4) did not consider the critical checkerboard mode, evidently due to the dierence in the periodicity wavelengths in the two sets o axes.

12 called hexagonal mode is a periodic mix o hexagons and triangles in the orm o a Kagome pattern. An alternative expression or (4.4) that more clearly reveals the triangular pattern is w 4 t sin kx sin kxsin kx (4.5) with x ( x 3 x)/ and x ( x 3 x)/. The three sets o nodal lines in (4.5) orm the triangular grid in Fig. 7. Eq. (4.5) and (4.4) are identical, as readily shown by trigonometric identities. Furthermore, they can be expressed as the sum o three critical D modes as w t( sin( kx ) sin( kx) sin( kx)). When a lat ilm orms a pattern o wrinkles, a network o lines exists separating regions where the ilm delects up or down. These lines are known as the nodal lines. In passing, it can be noted that the only periodic patterns o same-sized, regular polygons that can orm the nodal lines or periodic buckle modes are squares and equilateral triangles. This assertion ollows rom the act that these two shapes and the hexagon are the only same-sized regular polygons that can tessellate a plane. Note rom Fig. 7 that the square and the triangular modes have alternating signs in neighboring cells, consistent with the pattern edges being nodal lines. This is not possible or a pattern made up entirely o hexagons because three hexagons come together at each corner. The delection in the hexagonal pattern (4.3) depicted in Fig. 7 has the same sign at the center o all the hexagons and the opposite sign in all the triangles. The nodal lines are closed loops within each o the hexagons. The importance o the asymmetry in the delection o the hexagonal mode (4.3) was noted in connection with the experiments. A herringbone, or zigzag, mode will also be considered here. This pattern is not associated with the critical stress. To motivate the mode shape employed, consider vertical delections o the ilm in a herringbone pattern having a wavy ridgeline with uniorm height, t, and wavelength, / k, in the x direction: wtcos kx acos( kx) (4.6) ( t/ ) cos kx cos acos( kx) sin kx sin acos( kx) For a, this delection is well approximated by

13 Herringbone mode: w t cos( kx ) sin( kx )cos( kx ) (4.7) with / and a /. In the upper-bound analysis in the next section, the vertical ilm delection is taken as (4.7) with,, and treated a ree variables; k is speciied by (3.5). With and, Eq. (4.7) coincides with the D mode, and with and /, it coincides with the checkerboard mode. In general, however, (4.7) is not a combination o critical modes. 5. Upper-bound analysis o energy in buckled state The exact inite amplitude solution or the D mode will be quoted and used as a reerence. This is ollowed by the upper-bound analysis o the energy in the buckled state or the hexagonal pattern, which is given in suicient detail to reveal all the essentials o the analysis. Then the results o the upper-bound analyses or the other modes identiied in ection 4 will presented and discussed. (5.) The exact D solution The nonlinear equations, (3.)-(3.3), admit an exact solution (hen & Hutchinson, 4) or D modes o the orm w tcos( kx ) with as a wavelength actor which is unity or the critical eigenmode. With as the stress o the unbuckled ilm, the elastic energy/area o the ilm/substrate system in the unbuckled state is U E t (5.) The amplitude o the buckling delection is (5.) and the normalized elastic energy/area o the system in the buckled state, U, (evaluated over one ull buckle wavelength), is

14 U U 3 (5.3) As can be seen immediately, the minimum energy in the buckled state or the D mode is given by, corresponding to the critical mode at the onset o buckling, at all values o overstress, /. The normalized energy with is plotted in Figs. 8 and 9. (5.) Upper-bound analysis or the hexagonal mode The analysis o the energy in the buckled state will be presented in enough detail to bring out the essential analytical eatures. The approach was irst used by von Karman and Tsien (94) in their attempt to expose the nonlinear buckling behavior o cylindrical shells under axial compression. The analysis is similar in some respects to that employed by Audoly & Boudaoud (9a) but diers in that it makes no truncation in the evaluation o energy. In particular, emphasis here is placed on the upper-bound character o the results, and it will be shown that these estimates can be accurate to surprisingly large overstress. As the starting point, consider the generalization o the hexagonal mode (4.3): w t cos( kx ) cos( kx / )cos( 3 kx / ) (5.4) where and are independent amplitude actors and scales the size o the pattern. (It will be seen that minimizing the energy o the system gives and or all, thereby establishing that the critical hexagonal mode (4.3) is indeed preerred among all modes o the orm (5.4).) The irst step is to solve the compatibility equation (3.) or F exactly in terms o w in (5.4). 3 This ensures that in-plane displacements exist associated with the resultant in-plane stresses, and it provides the basis or arguing that the elastic energy o the system will be an upper-bound, i.e., the energy is computed rom admissible displacement ields. The result is This equation is obtained rom Eq. () o hen and Hutchinson (4). Eq. (3) o that paper, which has been specialized to, has a misprint: ( ) in the last term should be ( ). 3 Audoly and Boudaoud (8a) work directly with the in-plane displacements o the plate rather than the stress unction. By doing so, they do not need to enorce the compatibility conditions on the average inplane displacements (3.3). However, use o the stress unction signiicantly simpliies the calculations by reducing algebraic complication especially when all the contributions are retained in evaluating the energy. 3

15 3 3 F tx tx E t cos( kx) cos( 3 kx) 3 9 (5.5) 3 3 E t cos( kx/ )cos( 3 kx / ) cos(3 kx/ )cos( 3 kx / ) 8 9 The terms and are the reduction in the average in-plane stresses in the buckled state; they are obtained rom conditions (3.3) as E ( tk) 3 E ( tk), 4( ) 8 3( ) (5.6) The three contributions to the average energy/area o the system (plate bending and stretching, and substrate deormation, respectively) are D U ( w) ( )( w, w, w, ) d ( F) ( )( F, F, F, ) d Et i * E 4 k k cos ( k x )cos ( k x ) d () i () i () i () i i (5.7) where is the area o the periodic cell. The contribution rom the substrate is valid or any combination o modes o the orm w cos( k x ) cos( k x ). For the results in i () i () i (5.4)-(5.6), the energy/area can be evaluated without approximation as U ( ) t/ E (5.8) Here, ( ), ( ), / ( /8)/( ) and / 3 /[8( )]. Minimizing the energy with respect to (,, ) is equivalent to minimizing with respect to (,, ). One sees immediately that the minimum o U requires or all. It is then straightorward to show that the equations or the minimum o U with respect to and are linear with solution 5 4 and (5.9) 4 4

16 It ollows rom (5.6) (with ) that 6 5 (5.) The normalized energy/area evaluated using (5.9) and (5.) with is U U 5( ) 5 (5.) As emphasized previously, (5.) provides an upper-bound to the energy in the buckled state or all hexagonal patterns by virtue o the act that the energy is evaluated without approximation using admissible displacement ields. Among the hexagonal modes, the critical mode with gives the minimum energy at all overstresses (Figs. 8 and 9). (5.3) Upper-bound or the checkerboard mode The upper-bound to the energy/area or any rectangular checkerboard modes o the orm (3.5) is a special case o the analysis o the herringbone mode given in the next subsection. The ollowing indings emerge. First, among all the critical rectangular modes o the orm (3.5) the square checkerboard mode has the lowest energy in the buckled state. econdly, among all sizes o square checkerboard patterns, the critical mode (4.) is ound to give the lowest energy/area or all. For this mode, (3 )( ) (5.) and U U ( ) 3 with the latter included in Figs. 8 and 9. (5.3) (5.4) Upper-bound or equilateral triangular modes The upper-bound analysis carried out using (4.4) or the critical triangular mode gives precisely the same result or the energy in the buckled state, (5.), as or the hexagonal mode. The buckling amplitude is related to the overstress by 5

17 6 5 (5.4) In addition, one can extend the analysis by considering equilateral triangular patterns o any size and show that the upper-bound analysis predicts that the critical mode (4.4) gives the lowest energy in the buckled state. Interaction o triangular and hexagonal modes will be discussed in ection 6. (5.5) Behavior o critical modes at small overstress It is useul to compare the expansion o U / U or each the our critical modes or small overstress with / slightly above : U U q... q (D mode) q (square checkerboard mode) 3 (5.5) 6 q 5 (hexagonal and triangular modes) To order ( / ), these results are exact or the model. This ollows or the D mode because (5.3) is exact or the entire range o overstress. For the other modes, the exactness o these expansions to this order ollows rom the act that the upper-bound solution procedure is equivalent to a rigorous initial post-buckling analysis to this order as laid out by Koiter (945, 9). We have independently veriied that Koiter s initial post-buckling analysis yields the above results. As established by Audoly and Boudaoud (8a), the checkerboard mode has the lowest energy in the range o small overstress. The other three modes having nearly equal energies the D mode has slightly lower energy than the hexagonal and triangular modes i / 5 and vise versa. Fig. 8 suggests that the same ordering o the energies in the buckled state persists to large overstresses. The ull numerical simulations in 6

18 ection 8 bear this out with the exception that the D mode has the highest energy or / 3. Accompanying the asymptotic expression or the energy in the buckled state is the asymptotic relation between the mode amplitude and / : / (5.6) b where b can be identiied rom (5.), (5.), (5.) and (5.4): b (D mode) (3 )( ) b (square checkerboard mode) b (hexagonal and triangular modes) 4 (5.6) Upper-bound or herringbone modes The solution procedure based on the herringbone mode (4.7) is the same as that presented or the hexagonal mode. The stress unction is obtained exactly rom (3.) in terms o (4.7) as F tx tx 3 E t cos( kx) cos( kx) 3 ( ) sin( )cos( ) onditions (3.3) give 3 E t kx kx ( ) E ( kt) 8( ),, (5.7) (5.8) The elastic energy in the buckled state is evaluated without approximation using w and F with the result 7

19 U ( ) t/ E 4 (5.9) ( ) 6 ( ) ( ) now with ( ), ( ) and,, ( ) As remarked earlier, this result specializes to the exact D solution (5.3) when minimized with respect to and with. It has been used to generate the upperbound (5.) or the square checkerboard mode by taking and ollowed by minimization o U with respect to and. While U is quadratic in and, its dependence on and is too complicated to acilitate a closed orm solution or the minimum o U when both and are permitted to be nonzero. Nevertheless, it is straightorward to generate a minimum or speciied values o / using standard numerical minimization algorithms, as has been done or the upperbounds to U / U or the herringbone mode plotted in Figs. 8 and Mode transitions and equi-energy mode combinations According to the upper-bound results presented in Fig. 8, the herringbone mode has the lowest energy o all the periodic modes considered in this paper at suiciently large overstresses. This will be borne out by the numerical analysis present in ection 8. Nevertheless, because the herringbone mode (3.9) (with both and nonzero) is not a combination o the critical eigenmodes, there must be some range o / near unity where the critical modes have lower energy. This is brought out most clearly in Fig. 9 which is restricted to small to moderate overstress. With / 3, the herringbone 8

20 mode has the lowest energy when /.476 but in the range, /.476, the square checkerboard mode has the lowest energy o all the modes considered. The herringbone mode arises as a biurcation rom the square checkerboard mode. With / and minimizing the energy in (5.9) with respect to and, one inds that biurcation rom the square mode ( and b / rom (5.6)) into the herringbone mode with occurs when 5(3 ) (6 ) 4 3( ) 6 5(3 ) which gives /.357 or v ; /.476 or v /3; and (6.) /.595 or v /. This biurcation stress is not exact because the solution or the square mode is only exact to order. However, it should be reasonably accurate because the upper-bound analysis or the square mode is accurate to relatively large values o the buckling amplitude, as the numerical analysis in ection 8 reveals. An exact biurcation analysis rom the D mode was presented by Audoly and Boudaoud (8 a). The D mode is never the mode with the lowest energy when the undamental stress state is equi-biaxial compression. However, or undamental stress states that are not equi-biaxial, Audoly and Boudaoud present a map that deines the domain o states wherein the D mode is the preerred mode with stress limits deining the biurcation transitions to other modes including the herringbone mode. It has been noted in the previous section that the triangular mode and the hexagonal mode have precisely the same energy in the buckled state at all values o /, according to the upper-bound analysis. With this in mind, we have investigated whether linear combinations o these two critical modes exist, i.e. w t sin( kx ) cos( kx / )cos( 3 kx / ) T t cos( kx ) cos( kx / )cos( 3 kx / ), H (6.) with the same energy. Here, to produce the correct coupling between the modes, the x coordinate in the triangular mode in (4.4) has been shited: x x / k. The upperbound analysis can be carried out exactly with only slightly more algebraic complication 9

21 than that or the individual modes; the details will not be presented here. Remarkably, one inds that all mode combinations satisying 4 T H 6 5 (6.3) have the same energy, (5.), while all other combinations have higher energy. According to the upper-bound analysis, at all levels o overstress, there exists a continuous mode transition connecting the triangular and hexagonal modes along which the energy is constant. The transitional mode shapes along this path are displayed in Fig. plotted as contour plots using (6.) or pairs (, ) satisying (6.3). Another example o a continuous mode transition under increasing overstress will be presented in ection 8. T H 7. Role o initial ilm curvature on the sign o the delection and mode selection As anticipated earlier, several aspects o the results obtained so ar appear to be at odds with the experimental observations in ection. In particular, the hexagonal pattern is much more likely to be observed than the square checkerboard pattern in the range o low overstress even though the square pattern has a lower energy. To our knowledge, the triangular pattern has never been observed even though it has identical energy to the hexagonal pattern, to the accuracy determined in this paper. In addition, in all instances in which the hexagonal pattern has been observed by us, the regions interior to the hexagons always delect into the substrate (i.e. in (4.3)). It is straightorward to show that model speciied in ection 3 implies that the solutions are independent o the sign o the normal delection. That is, i ( wf, ) is a solution to the governing equations, then so is ( wf, ). The numerical inite element results o the 3D model with ull coupling to the substrate presented in the next section also relect this symmetry, i not exactly, at least to high approximation. As described in ection, the swelling o the ilm/substrate system produces a slight initial curvature o the ilm such that the substrate is on the concave side o the surace. In this section, the analysis presented in ection 5 will be extended to account or an initial curvature o the ilm by modeling it as a shallow spherical shell.

22 Let R be the radius o curvature o the shallow spherical shell segment representing the ilm deined positive with the concave side down. The model in ection 3 is modiied by accounting or the curvature o the ilm. The eect o the curvature o the substrate on the restoring orce on the ilm, p, is ignored under the assumption that the shell eect is dominant. The modiied nonlinear equations are known as the Donnell- Mushtari-Vlassov shell equations and are equivalent to the shallow shell equations; they reduce to the Karman plate equations as R. The equations are modiied as ollows: R F is added to the let hand side o Eq. (3.), side o Eq. (3.), and R w R w is added to the right hand is added to the integrand o the integral on right hand side o both equations o (3.3). The expression or the energy o the system in (5.7) still holds. onsider modes o the orm cos w t cos kx kx (7.) with and * as ree variables and /3 k t 3 E / E as beore. These modes are restricted to a shallow segment o the shell, consistent with the act that the wavelengths are very small compared to R. The coordinates ( x, x ) are local orthogonal coordinates in the middle surace o the shallow shell segment. R The biurcation analysis gives the ollowing equations or the critical stress, : R 3 3 with 4 3 (7.) and. 4 Here, is still (3.4), and the dimensionless curvature parameter is t E ( ) * R 3E /3 (7.3) 4 A nonlinear analysis o buckling o an unsupported spherical shell subject to external pressure was carried out by Hutchinson (967) based on the same equations. The normalizations in (7.) and (7.) break * R down in the limit when E, but one inds the critical stress E ( t/[ 3( ) R]) associated with a similar amily o short wavelength modes. The hexagonal pattern is the most unstable. A recent review o buckle patterns on curved compliant substrates o various shapes has been given by hen and Yin ().

23 As in the case o the lat ilm, all modes o the orm (7.) with are associated with the critical stress. For, as is the case or the experiments in R ection, the above equations give / and to order. In the remainder o this section, the upper-bound analysis or the hexagonal mode, 3 w t coskxcos kx cos kx, (7.4) is presented based on an analysis similar to that in ection 5. In this analysis, is ixed at the value given in (7.). The stress unction is t F x x E t ( cos( kx) cos( kx/ )cos( 3 kx) 8 4 cos( 3 kx / ) cos(3 kx/ ) cos( 3 kx / ) ) 4 Et cos( kx) cos( kx/ )cos( 3 kx) R k (7.5) with (3 / )( ). The strain energy rom (5.7) is U 3 ( ) ( ) 3( ) R t/ E 9 5 ( ) ( ) 8 3/ As in the upper-bound analysis o the lat ilms, no approximation has been made in arriving at (7.6) given (7.4) as the starting point. Minimizing U with respect to gives 9 ( ) which coincides with (5.) when the curvature vanishes. The curvature introduces asymmetry in the buckling response as is evident in the plot o (7.7) in Fig.. With (7.6) (7.7) R, the applied stress,, initially diminishes ater biurcation when the buckling mode amplitude,, is negative. The biurcation point is unstable, and the system is highly sensitive to imperections that push it towards negative (Koiter, 945; 9). This asymmetry is due to shell curvature not the placement o the substrate. The

24 implication o (7.7) is that there is a preerence or hexagonal regions to buckle towards the concave side o the ilm inward as opposed to outward in the ollowing discussion. According to the model, this would still be true i the substrate were attached to the convex side o the ilm. The normalized energy in the buckled state, U / U, computed as unction o overstress, / R, using (7.6) and (7.7), is plotted in Fig.. The energy associated with inward buckling o the hexagonal regions is signiicantly lower than that associated with outward buckling. It should be noted that the plot in Fig. has been limited to states with / R. The energy or / R when is only very slightly below unity or the values o considered in Fig., as can be inerred rom the intercepts at / R. The asymmetry in buckling response with respect to the amplitude o the hexagonal mode does not occur or either the triangular mode or the square checkerboard mode because the opposite-signed regions o these biurcation modes are identical. More importantly, the upper-bound calculation or the square mode (details omitted) reveals that the normalized relationship, U / U versus / R, is independent o. For the square mode, w tcos( kx/ )cos( kx / ), (7.) applies and U / U is given by (5.3) with / replaced by / R. The question o whether a slight curvature can account or the preerence or the hexagonal mode over the square mode in the range o small overstress can now be addressed based on the assumption that the mode with the lowest energy is the most likely to be observed. For speciied a value o / R, Fig. 3 plots the value o or which the energy o the hexagonal mode (buckling inward) equals that o the square mode. For to the let o the curve the square mode is preerred while to the right o the curve the hexagonal mode is preerred. The value o required to avor the hexagonal mode over the square mode is very small or very small overstress, approaching zero in the limit o zero overstress. Thus, any initial curvature would tend to select the hexagonal mode or a perect system subject to increasing stress such that / R exceeds unity rom below. Initial imperections will play some role in promoting 3

25 individual modes. Imperections are also likely to avor hexagonal over square patterns in the presence o curvature because buckling inward in the hexagonal mode is unstable and strongly imperection-sensitive, whereas the square mode is not imperectionsensitive. Imperection-sensitivity causes the amplitude o the hexagonal mode to be ampliied more than the others, assuming there exists an imperection component in the shape o the mode. The observed hexagonal mode in Fig. 3 is ully developed at / R.3. It is reasonable to assume that this mode has been selected or / R.. Based on Fig. 3, curvature would ensure the selection o the hexagonal mode under this circumstance i.5. To estimate the implication or the experimental system described in ection, re-express using the wavelength o the hexagonal mode, L 4 /( 3 k) : 3 3 L L R (7.8) 8 tr 8 t With.5, L 5m (c.. Fig. ) and t nm, one inds R 4mm. For a specimen width mm, this curvature would imply a rise o.3mm at the center o the specimen relative to its edges. I, instead, one assumed the hexagonal mode was selected at / R. such that., then (7.8) gives R mm and the relative rise at the center o the specimen would be.3mm. Direct measurements o the initial curvature o the specimens have not been made, however, the above curvature estimates are not inconsistent with observations made by eye. Thus, tentatively, we have concluded that curvature is the source o the preerence or the hexagonal mode to buckle inward and or the hexagonal mode to be avored over the square mode. Initial curvature o the system would also explain why the triangular mode has not been observed. 8. Numerical analysis o nonlinear buckling o periodic patterns The inite element code, ABAQU ( has been used to carry out three dimensional analyses or all the patterns described in the ection 4 or ilms that are lat in the unbuckled state. For each o the patterns, a periodic cell is identiied. Both the ilm and the substrate are meshed with 8 node linear block elements with reduced integration. The ilm is represented by 8 layers o elements with approximately,5 4

26 elements in each layer o the cell. The substrate has depth d which is taken to be ive times the largest periodicity wavelength and its bottom surace is prescribed to have zero displacement. The substrate is represented by approximately, elements uniormly distributed in layers. The material or ilm and substrate are taken to be described by linear isotropic elasticity, except or a ew simulations or the hexagonal pattern where a neo-hookean substrate has been assumed to explore whether substrate nonlinearity is important. The nonlinear geometric updating option o ABAQU is employed. tress in the ilm is induced by a thermal expansion mismatch with the substrate by imposing a temperature drop T starting rom the unstressed state. For this purpose, the ilm is assigned a thermal expansion coeicient while the substrate is assumed to have zero thermal expansion. For all calculations, E / E 3465 with / 3 and.48. For each mode, periodic boundary conditions are imposed on the edges o the ilm/substrate cell. In all cases, the cell dimensions are set by the wavelengths associated with the critical modes introduced in ection 4, consistent with the act that the upperbound analysis suggests these produce the lowest energy states even at inite delection. A very small initial geometric imperection in the orm o a stress-ree out-o-latness in the shape o the buckling mode is used to promote the mode. peciically, with any mode in ection 4 denoted by w t ( x, x), the initial imperection or that mode is taken as w t ( x, x) with set at.. The D mode was simulated as a check on accuracy given the availability o the exact solution o the model problem or this case. A rectangular parallelepiped unit cell o dimension LL d was employed with L / k and d 5L. As the initial imperection has variation in only one direction, the induced mode varies only in that direction. The mode shape is projected over many cells and plotted in Fig. a. The normalized energy in the buckled state is plotted in Fig. 4. It agrees with the exact solution or the model in (5.3) (with ) to within one or, at most, two percent over the entire range plotted. The square checkerboard mode uses the same unit cell as the D mode except that L / k and the imperection is in the shape o the square mode. In this case the square mode is generated (Fig. b) and the normalized energy is plotted in Fig. 4. 5

27 The numerical result is very close to the upper-bound prediction or U / U in Fig. 8 or / 4 and alls no more than 3% below the upper-bound or / as large as. The unit cells or the hexagonal mode and the equilateral triangular mode are hexagonal prisms as depicted in outline in Fig. 7. For the hexagonal mode the distance across the cell rom ace to ace is 4 / 3k while the ace to ace distance o the cell or the triangular mode is 4 /k. An imperection in the shape o the mode is introduced or each calculation as described above. In addition, or the hexagonal mode, calculations with both positive (hexagon centers delecting outwards, ) and negative ( ) imperections are preormed. The mode shapes are shown in Fig. and the results or U / U are plotted in Fig. 4. There is virtually no dierence between the energy in the buckled state or inward and outward delections o the hexagons, although there are slight dierences in the delections themselves when / is large. Unlike the plate model, the 3D model o the system is not strictly independent o the sign o the normal delection o the ilm, but it is evident rom Fig. 4 that the dierence is extremely small or the overstress range considered. imilarly, as the upper-bounds or the plate model predict, Fig. 4 reveals that there is virtually no dierence between U / U or the triangular and hexagonal modes. The upper-bound is quite accurate or / 3 and then begins to diverge or larger overstress; it becomes increasingly inaccurate or larger overstress with U / U.4 rom the numerical analysis and U / U.48 rom the bound at /. Nevertheless, it is quite remarkable that the bound is highly accurate to overstresses as large as three times the critical stress. An intriguing mode transition occurred in the numerical simulation or the triangular mode which is displayed in Fig. 5. As expected, the delection initially assumes the shape o the triangular pattern as the overstress is increased due to the assumed initial imperection, but at a certain level o overstress ( /.4 in Fig. 5) the mode begins to develop an asymmetry not present in the initial mode. When / 3.4 the transition to a three-lobed asymmetric pattern is ully developed, and new mode persists as the overstress is urther increased. A transition rom the triangular mode to the hexagonal mode did not occur in these simulations because the size o the 6

28 computational cell or the triangular mode was larger by a actor 3 than that or the hexagonal cell. One other set o inite element calculations or the hexagonal mode was carried out to gain urther insight into the observed preerence or this mode to buckle into the substrate at the centers o the hexagons. Recent work (Brau et al., ; un et al., ) has shown that the nonlinearity o the substrate aects the mode shape at very large overstress when the slopes o the ilm become large. In the present calculations, the neo- Hookean material is used to represent the substrate. Positive and negative imperections were considered so as to promote both inward and outward delections o the hexagonal model. The substrate material (the neo-hookean option in ABAQU) was taken to have the same slight compressibility (.48 ) and modulus ( E / E 3465) in the ground state as the linear material used in all the other calculations. The results o these calculations are shown in Fig. 6 where it is seen that they are essentially indistinguishable rom the predictions based on the linear substrate. Even at / the delection o the ilm is still too small to push the substrate into the nonlinear range. I E / E, it is easy to show that the maximum slopes o the ilm are only on the order o / or /. Thus, the period doubling seen by Brau et al. () and un et al. () and the hierarchical wrinkling seen by Eimenko et al. (5) are not expected in the range o overstress considered in this paper. An extensive numerical analysis o the herringbone mode or a 3D model which uses plate theory to model the ilm was conducted by hen and Hutchinson (4). The reader is reerred to that study or details o the cell geometry and how that geometry was varied to produce herringbone modes with minimum energy in the buckled state. Here we have employed the parameters o the cell ound by hen and Hutchinson to give the minimum, or near-minimum, energy a cell with width L / k perpendicular to the ridge lines and with a ridge kink angle o 9 o. The pattern is displayed in Fig. e, and U / U is plotted in Fig. 4. As the upper-bound analysis had predicted, the herringbone mode has the lowest energy in the buckled state among all the modes considered or all / greater than about.5. The upper-bound is less accurate or the herringbone 7

29 mode than the other modes, no doubt due to the act that the assumed delection in (4.7) is itsel an approximation to a herringbone mode. Thus, at / 3 the bound is about % higher than the numerical result and at / it is more than 3% too high. In summary, the numerical analysis o the 3D model o the ilm/substrate system indicates: (i) The present results or the square and herringbone modes are almost identical as those ound by hen and Hutchinson (4) or a 3D model in which the ilm is modeled by plate theory. (ii) With the exception o the herringbone mode, the upperbound results in ection 4 are reasonably accurate to surprisingly large overstresses as large as three or our times. (iii) The models or initially lat ilms, including the incorporation o nonlinearity in the substrate, do not indicate any appreciable dierence between inward and outward buckling o the hexagonal mode, nor is substrate nonlinearity important or the range o overstress considered in this paper. 9. oncluding remarks Many details o the energy landscape in the buckled state o the periodic modes considered here have been revealed by making use o the analytical upper-bound method in combination with the 3D inite element analysis or systems that are lat in the unbuckled state. The upper-bound retains its accuracy to surprisingly large overstresses or the critical modes. Based on the notion that the mode with the minimum energy is the most likely to be observed, the square mode should be the dominant mode or overstresses, /, below about.5 and the herringbone mode should emerge at overstresses above that. The herringbone mode does begin to emerge in our experiments or /.5, but the observed mode below this is the hexagonal mode not the square mode. Moreover, in all cases in which we have observed the hexagonal mode, it buckles with the hexagonal regions pushing into the substrate whereas theory suggests there should be no preerence. We have established that a slight initial curvature o the ilm could explain the two discrepancies just noted and the act that the triangular mode has not been observed in our experiments even though it has the same energy as the hexagonal mode when the system is initially lat. The magnitude o the initial curvature observed in our system appears to be consistent with the curvature level needed to 8

30 explain the discrepancies but quantitative measurements necessary to establish this as a certainty have not been made. As just noted, or lat ilm/substrate systems, theory suggests that the square mode should be observed in the range o low overstress and it should give way to the herringbone mode with increasing overstress. However, in the range o low overstress there is a rich array o competing modes with energies that are only slightly separated. Mode selection based on minimum energy considerations has its limitations. The attempt initiated by von Karman and Tsien (94) to understand the post-buckling and collapse o cylindrical shells under axial compression provides an illustration o the limitations. Following the work o von Karman and Tsien, many papers were written aimed at more and more accurate evaluation o the energy in the collapsed state. Nevertheless, no clear understanding emerged as to the relevance o minimum energy states to the load carrying capacity and collapse behavior o the shell. It was not until Koiter (945) emphasized the importance o initial imperections and included them in his approach that the highly unstable buckling behavior o cylindrical shells under axial compression was related to the strong imperection-sensitivity o the maximum load carrying capacity. Unlike cylindrical shells, the ilm wrinkling problem or lat ilms has a stable buckling behavior and is not imperection-sensitive in the way the shell is. Nevertheless, initial imperections may be playing a role in mode selection given the small dierence in energy between the modes when the overstress is small. I imperections are included in the analysis, as Koiter (945, 9) has done in his general theory, the development o a speciic mode is promoted by imperections in the shape o that mode. Once a periodic mode pattern develops there is a barrier to its evolving to another periodic mode with dierent wavelengths even i the other mode has lower energy. everal examples where transitions rom one mode to another occur continuously without a jump in wavelengths were presented in the paper, including the transition rom the triangular to the hexagonal mode and the experimental observation o the hexagonal mode evolving into a herringbone mode. Biurcation rom the square mode into the herringbone mode is another example. In conclusion, this paper has shed urther light on the wrinkling behavior o ilms under equal biaxial stress states. It has also exposed a number o open questions highlighting the richness o the nonlinear buckling phenomena. 9

31 Acknowledgements. ai, Z. uo and J.W. Hutchinson acknowledge the support o the MRE at Harvard.. ai and Z. uo also acknowledge the support o National cience Foundation on ot Active Materials (MMI-86). D. Breid and A. rosby acknowledge the inancial support o the National cience Foundation Materials Research cience and Engineering enter (MRE) on Polymers at the University o Massachusetts Amherst (NF-DMR 856 ). 3

32 Reerences Allen, H. G., 969, Analysis and design o sandwich panels. Pergamon Press, New York. Audoly, B, Boudaoud, A., 8a, Buckling o a thin ilm bound to a compliant substrate Part I: Formulation, linear stability o cylindrical patterns, secondary biurcations. J. Mech. Phys. olids 56, 4-4. Audoly, B, Boudaoud, A., 8b, Buckling o a thin ilm bound to a compliant substrate Part II: A global scenario or the ormation o herringbone pattern. J. Mech. Phys. olids 56, Audoly, B, Boudaoud, A., 8c, Buckling o a thin ilm bound to a compliant substrate Part III: Herringbone solutions at large buckling parameter. J. Mech. Phys. olids 56, Biot, M.A., 965, Mechanics o incremental deormation. Wiley, New York. Bowden, N., Brittain,., Evans, A.G., Hutchinson, J.W., Whitesides, G.M., 988. pontaneous ormation o ordered structures in thin ilms o metals supported on an elastomeric polymer. Nature 393, Brau, F., Vandeparre, H., abbah, A., Poulard,., Boudaoud, A., Damman, P.,. Multiple-length-scale elastic instability mimics parametric resonance o nonlinear oscillators. To be published in Nature Materials. Breid, D., rosby, A.J., 9. urace wrinkling behavior o inite circular plates. ot Matter 5, Breid, D., rosby, A.J.. in preparation or publication. han, E.P., rosby, A.J., 6. pontaneous ormation o stable aligned wrinkling patterns. ot Matter,, hen, X., Hutchinson, J.W., 4. Herringbone buckling patterns o compressed thin ilms on compliant substrates. J. Appl. Mech. 7, hen, X., Yin, J.,. Buckling patterns o thin ilms on curved compliant substrates with applications to morphogenesis and three-dimensional micro-abrication. ot Matter, 6, Eimenko, K., Wallace, W.E., Genzer, J.,. urace modiication o sylgard-84 poly(dimethyl siloxane) network by ultraviolet and ultraviolet/ozone treatment. J. olloid Interace ci. 54,

33 Eimenko, K., Rackaitis, M., Manias, E., Vaziri, A., Mahadevan, L., Genzer, J., 5. Nested sel-similar wrinkling patterns in skins. Nature Materials, 4, Hill, R., Hutchinson, J. W., 975. Biurcation phenomena in the plane tension test. J. Mech. Phys. olids 3, Huang, Z.Y. Hong, W., uo, Z., 5. Nonlinear analyses o wrinkles in a ilm bonded to a compliant substrate. J. Mech. Phys. olids 53, -8. Huang, R., 5. Kinetic wrinkling o an elastic ilm on a viscoelastic substrate. J. Mech. Phys. olids 53, Huck, W., Bowden, N., Onck, P., Pardoen, T., Hutchinson, J.W., Whitesides, G.M.,. Ordering o spontaneously ormed buckles on planar suraces. Langmuir 6, Hutchinson, J.W., 967. Imperection sensitivity o externally pressurized spherical shells. J. Appl. Mech Koiter, W.T., 945. On the stability o elastic equilibrium (in Dutch with English summary). Thesis Delt, H.J. Paris, Amsterdam. An English translation is available online: Koiter, W.T., 9. Elastic stability o solids and structures. A.M. van der Heijden (Ed.), ambridge University Press, ambridge. Lin, P.-., Yang,., 7. pontaneous ormation o one-dimensional ripples in transit to highly ordered two-dimensional herringbone structures through sequential and unequal biaxial mechanical stretching. Appl. Phys. Lett. 9, 493. Mills, K.L., Zhu, X., Takayama,., Thouless, M.D., 8. The mechanical properties o a surace-modiied layer on poly(dimethylsiloxane). J. Mater. Res. 3, un, J-Y., Xia,., Moon, M-Y., Oh, K.H., Kim, K-.,. Folding wrinkles o a thin sti layer on a sot substrate. submitted or publication. Von Karman, T., Tsien, H.., 94. The buckling o cylindrical shells under axial compression. J. Aero. ci. 8,

34 Appendix: Three plane strain analyses o the critical stress In linear elasticity the plane strain relation between sinusoidal tractions applied on the upper surace o a semi-ininite hal-space, (, ˆ ˆ ) ( p cos( cx ), t sin( cx )), and the associated displacements, ( u, u ˆ ˆ ) ( wcos( cx), usin( cx)), is pˆ wˆ ce ( ) ( ) ( ) B, B tˆ uˆ (3 4 ) ( ) ( ) (A.) von Karman plate representation o the ilm including tangential tractions Next consider a lat plate which in the unbuckled state is subject to an in-plane compressive stress. The von Karman strain-displacement relations and the Principle o Virtual Work are invoked, including downward normal and letward tangential applied tractions, ( pˆ cos( cx ˆ ), t sin( cx )). The linearized equations or perturbations o the middle surace displacements about the unbuckled state, ( wˆcos( cx ), uˆsin( cx )), are ( Dc tc) wˆ pˆ tct ˆ / Etcu 4 ˆ tˆ (A.) where i the tractions are imagined to be applied along the plate middle surace and i they are applied along the bottom surace o the plate. The eigenvalue problem or the critical value o is obtained by eliminating ˆp and ˆt by coupling (A.) and (A.). With c k where here kt (3 E / E ) /3, one obtains c cc c c c c ( E /4)(3 E / E ) 3 c 4c 3 /3 3 3 (A.3) with 4( ) c 3(3 4 ), c ( )( ) and 3(3 4 ) The critical value o, obtained as the minimum with respect to, has been plotted in Fig. 5 or / 3 or both and. Note that or these results, as or the other c 3 3E two estimates quoted in ection, the ilm moduli appear only through E /3 E. 33

35 The exact solution or the critical stress In this analysis, the ilm is taken to be a uniormly pre-stressed layer o thickness t that is attached along its bottom surace to the compliant substrate along the interace at x. The ilm layer is taken to be incompressible and undergoes incremental plane strain deormations. Incompressibility acilitates the exact biurcation analysis o the system using the ormulation o Biot (965) and Hill and Hutchinson (975). As noted earlier, the assumption o an incompressible ilm layer is not expected to reduce the applicability o the results since or all the other estimates only the plane strain modulus o the ilm, E, appears in the results. The ilm layer is coupled to the substrate which has no pre-stress and is governed by (A.). The reader is reerred to Hill and Hutchinson (975) or the general ormulation. Here only an outline o the results will be presented. A displacement potential, ( x, x) (/ c) W( x)sin( cx) is used to represent the perturbations o the displacements rom the uniorm pre-buckling state within the layer u (/ c) dw / dx sin( cx ), u W cos( cx ) (A.4) The general solution to the governing ield equation or is W( x) ae ae ae ae (A.5) cx cx dcx dcx 3 4 with d / E / / E. The perturbed tractions vanish on the upper surace o the layer at x t i where a a ce ce ( d) ct ( d) ct 3 B A M, M ( d ) ct ( d ) ct a a 4 ce A ce B (A.6) 3 c 3 A d d d 4 E E 3 c 3 B d d d 4 E E Both (A.6) and ( u, u ˆ ˆ ) ( wcos( cx), usin( cx)) along x will be satisied i A ( d ) ct ( d ) ct ( d ) ct ( d ) ct a ˆ 3 w ce B ce A ce A ce B, A ( d ) ct ( d ) ct ( d ) ct ( d ) ct a ˆ 4 u ce B ce A d ce A ce B d (A.7) 34

36 In the notation o Hill and Hutchinson (975), denote the nominal traction increments along the lower surace o the ilm layer by ( n ˆ ˆ, n ) ( pcos( cx ), tsin( cx )). These are the vertical and horizontal components o the orce per pre-biurcated area ollowing the surace as it delects. It is these nominal traction increments that must be continuous across the interace. They are connected to the displacement perturbations by pˆ wˆ E B, B c A tˆ uˆ (A.8) 4 with 3 3 A / E d d A / E d d A d A d Finally, the ilm layer is coupled to the substrate by imposing continuity o the increments o displacement and tractions using (A.) and (A.8): wˆ B B uˆ (A.9) As in the previous analysis, one can let c k with k given by (3.5). The equation or / E is det B B which, other than, depends only on two dimensionless parameters, E / E and ; must be varied to produce the lowest value o / E. The matrix inversion and multiplication giving B in (A.8) can be perormed analytically; the results in Figs. 5 and 6 have been computed based on B B det. It is worth noting that the exact result or the critical stress lies above the prediction (A.3) with because the von Karman plate equations neglect elastic energy stored in transverse (out-o-plane) shear, which is ully accounted or in the exact ormulation. 35

37 (a) (b) (c) (d) (e) Fig. chematics o mode shapes: (a) D mode, (b) quare checkerboard mode, (c) Hexagonal mode, (d) Triangular mode, (e) Herringbone mode. 36

38 a) b) c) d) Fig. Experimental observations o our buckling modes o ilms on PDM substrates: a) D mode, observed when one principal in-plane stress dominates; b) square checkerboard mode; c) hexagonal mode (inset: ast Fourier transorm o a selected "grain" o depressions, showing the hexagonal ordering); and d) herringbone mode. Images a, c, and d are optical micrographs o wrinkles generated via the swelling process outlined in ection. Image b depicts the surace proile o a lat polystyrene ilm on a PDM substrate, wrinkled due to thermal mismatch stresses. The checkerboard pattern in image b is not readily observed. In this case it occured within a relatively small region surrounded by other wrinkle patterns. 37

39 Fig. 3 Progression o modes observed experimentally or the UVO-treated PDM system with increasing overstress as described in the text. The UVO treatment times rom let to right are, 5,, 3, 45, and 6 minutes. Fig. 4 A sequence o pictures depicting the transition o the hexagonal mode to a segmented labyrinth (disorganized herringbone) pattern with increasing overstress. 38

40 Fig. 5 ritical stress or plane strain deormation o the ilm/substrate system with /3. The reerence is ( ) ( /4)(3 / ) /3 A E E E ; ( ) ( E /4)(3 E / E ) * /3 B with E 4( ) E / (3 4 ). ases (I), (II) and (III) are described in the text; * ( ) is exact. III 39

41 Fig. 6 The exact critical stress, ( ) or plane strain deormation o the ilm/substrate III system or various Poisson ratios o the substrate with ( ) ( /4)(3 / ) /3 A E E E. The ilm layer is incompressible. 4

42 Fig. 7 Three o the our critical buckling modes as deined in (4.)-(4.4). The top view o the computational cell or the hexagonal and triangular modes is outlined by heavy dashed lines. 4

43 Fig. 8 Normalized energies or the various modes in the buckled state according to the upper-bound analysis with /3. 4

44 Fig. 9 Normalized energies in the buckled state in the range o small to moderate overstress according the upper-bound analysis with /3. 43

45 Fig. Linear combinations o the triangular and hexagonal modes that have precisely the same energy according to the upper-bound analysis. This property holds or all values o overstress, /. 44

46 Fig. Eect o spherical curvature on the relation between the applied stress and the buckling amplitude or the hexagonal mode. The centers o the hexagon preer to buckle towards the concave (inward) side o the curved ilm. 45

47 Fig. Eect o curvature on the energy in the buckled states o the hexagonal mode. 46

48 Fig. 3 For a given the curve gives the value o or which the energy o the hexagonal mode (buckling inward) equals that o the square mode. A mode-selection criterion based on minimum energy avors the hexagonal mode to the right o the curve and the square mode to the let o the curve. 47

49 Fig. 4 Normalized energy in the buckled state or the various modes as determined rom the numerical analysis o the 3D models. 48

50 Fig. 5 Transition rom a triangular mode to a asymmetric three lobed mode under increasing overstress. The result was computed with the three-dimensional inite element model using the computational cell or the triangular mode. 49